What is different between maximal cyclic subgroup with cyclic maximal subgroup? What is different between  "maximal cyclic subgroup" with "cyclic maximal subgroup"?
Are these are equal or different?(example?)
 A: A maximal cyclic subgroup is a cyclic subgroup that is maximal among all cyclic subgroups. That is there is no other cyclic subgroup that would contain it while there might be other non-cyclic subgroups that are larger.  
A cyclic maximal subgroup is a maximal subgroup (among all subgroups) that is cyclic. That is it is a maximal subgroup that happens to be cyclic. 
I think this is the default reading, but I would not rely on everybody using or understanding it in this way.
Note that a cyclic maximal subgroup is always a maximal cyclic subgroup but the converse might not be true. 


*

*In the additive group  $\mathbb{Z}/p^2\mathbb{Z} \times \mathbb{Z}/p^3\mathbb{Z} $. 
The subgroup $ \{0 \} \times \mathbb{Z}/p^3\mathbb{Z}$ is a maximal cyclic subgroup, yet not a maximal subgroup and thus not a cyclic maximal subgroup either. 

*In the additive group  $\mathbb{Z}/p\mathbb{Z} \times \mathbb{Z}/p^3\mathbb{Z} $. 
The subgroup $ \{0 \} \times \mathbb{Z}/p^3\mathbb{Z}$ is a maximal cyclic subgroup, and also a maximal subgroup and thus a cyclic maximal subgroup.
